2
$\begingroup$

Is there a name for a position on a function surface and the value at that position? E.g., if I have the function $f(x,y) = x^2 + y^2$, and I know that at the point $(2, 2)$ it evaluates to $8$, is there a name for the data structure $(2, 2, 8)$ [which is in this example $(x, y, f(x, y))$]?

$\endgroup$
  • $\begingroup$ This doesn't quite answer your question, but the set of all points of the form $(x,y,f(x,y))$ is called the graph of the function $f$, so in your case, $(2,2,8)$ is a point in the graph. In general, for any function $f : X \to Y$ between two sets, the set $\{ (x,f(x)) \mid x \in X \} \subseteq X \times Y$ is the graph of $f$. $\endgroup$ – fuglede Oct 5 '11 at 20:36
2
$\begingroup$

It is an element of the graph of $f$. The formal definition of the graph of a function $f:X\to Y$, where $X$ and $Y$ are any two sets, is the subset $\Gamma(f)\subset X\times Y$ consisting of $$\Gamma(f)=\{(x,y)\in X\times Y\mid y=f(x)\}.$$ So, in your case, $X=\mathbb{R}^2$ is the plane, $Y=\mathbb{R}$ is the real numbers, and $$\Gamma(f)=\{((x,y),z)\in \mathbb{R}^2\times \mathbb{R}\mid z=f(x,y)\}.$$ But we usually identify $\mathbb{R}^2\times\mathbb{R}$ with $\mathbb{R}^3$, of course, and we get $$\Gamma(f)=\{(x,y,z)\in \mathbb{R}^3\mid z=f(x,y)\},$$ or in other words, $\Gamma(f)$ consists of the triples $(x,y,f(x,y))$, such as $(2,2,8)$.

$\endgroup$
  • $\begingroup$ Thanks. I am guessing there is no fancy name for "element of the graph of f"? $\endgroup$ – shino Oct 5 '11 at 21:24
  • $\begingroup$ No, I don't believe so. $\endgroup$ – Zev Chonoles Oct 6 '11 at 2:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.